- 359

- 3

If r is partial ordering on X, prove that r is contained in a total ordering on X. Hint: Consider the collection of all partial orderings containing r. Use Zorn's Lemma.

I've already proven using Zorn's Lemma that there exists a maximal partial ordering m containing r. But I can't seem to prove that m is a total ordering. Suppose a and b are not comparable by m. I tried to prove that m U {(a,b)} is a partial ordering (to contradict the maximality of m), but can't. What's the correct contradiction? How about I make b greater than every element in X (that is not already greater than b)? Nope that doesn't work either (transitivity fails).

I've already proven using Zorn's Lemma that there exists a maximal partial ordering m containing r. But I can't seem to prove that m is a total ordering. Suppose a and b are not comparable by m. I tried to prove that m U {(a,b)} is a partial ordering (to contradict the maximality of m), but can't. What's the correct contradiction? How about I make b greater than every element in X (that is not already greater than b)? Nope that doesn't work either (transitivity fails).

Last edited: